FIG. 1 schematically shows an optical imaging system. Simplified to the minimum, this imaging system includes an optical system 10 and an image sensor 30. In the example shown, the optical system 10 is consisted of a single lens with a symmetry of revolution about an optical axis 12. The optical system 10 may include an assembly of several lenses or other optical components. A pupil 11 delimits the aperture of the optical system 10. Generally, the centre O of the pupil 11 is placed on the optical axis 12. The image sensor 30 comprises at least one elementary sensor 31, 32, 33. By elementary sensor, it is understood herein a sensor having a resolving element in a least one spatial direction. The image sensor 30 is arranged in a sensing plane 40, generally transverse to the optical axis 12. A light source (not shown) illuminates an object located in the object space 200. The optical system 10 forms an image of this object in the image space 400. More precisely, the optical system 10 optically conjugates the sensing plane 40 with an object plane 20. The focusing depth PM of the imaging system is defined as being the longitudinal interval of the image space 400 for which a given image sensing system 30 senses no or a little blurring in the sensed image. In the object space 200, the depth of field PDC of the imaging system is defined based on the focusing depth PM by applying the usual relations of optical conjugations. The focusing depth PM depends on the specifications of the optical system (focus, object-image optical conjugation, numerical aperture), on the wavelength λ of the optical beam and on the spatial extension of the resolving element of the image sensor 30.
Schematically, it has been shown in FIG. 1 a sensing system 30 formed, for example, of a planar matrix of elementary sensors 31, 32, 33, or pixels (CCD or CMOS), a strip of pixels or a single sensor (photodiode), of predetermined size. The pixel 31 is arranged in the sensing plane 40, at the intersection with the optical axis 12. The pixel 31 senses the image A′0 of a point A0 located at the intersection of the optical axis 12 with the focusing plane 20. The optical system 10 optically conjugates the focusing plane 20 of the object space with the sensing plane 40 of the image space. Hence, the sensing plane 40 is a plane transverse to the optical axis 12 and optically conjugated with the object plane 20. Similarly, a point Ap, respectively Ar, located in an object plane 21, respectively 22, is optically conjugated with a point A′p, respectively A′r, located in an image plane 41, respectively 42. By definition, the points Ar and A′p corresponding to the ends of the geometrically calculated axial area of focusing depth PM, the transverse size of the optical beam forming the image at the point Ar and A′p has a size equal to the size of the pixel 31, in the sensing plane 40. Hence, the sensing system senses no degradation of the image quality between the object planes 20, 21 and 22. The depth of field PDC geometrically corresponds to the distance along the optical axis 12, between the object plane 21 and the object plane 22 for with the image through the optical system 10 has the transverse size of one pixel 33 of the sensing system 30.
A simple geometric analysis shows that the focusing depth PM depends on parameters such as in particular: the numerical aperture of the optical system, the characteristic size of the elementary sensor and the magnification of the optical system.
Different techniques exist for increasing the depth of field of an optical imaging system. Apodizer-type spatial filters are known, which are placed in the plane of the optical system pupil. An apodizer filter is generally a transmission spatial filter having an amplitude distribution that allows suppressing the secondary rings of the diffraction pattern produced by an optical system at one image point.
In FIG. 2 are shown the geometric reference systems used for evaluating the amplitude of the impulse response in the image field, near the focusing plane, of an optical system having a given pupil function 15.
An imaging system is considered, comprising an optical system of focal length F forming the image of a point A0 at an image point A′0. The imaging system further includes a spatial filter or optical mask. This optical mask is arranged near the plane of the optical system pupil or in a plane optically conjugated with the plane of the optical system pupil. The optical mask generates a complex pupil function 15 liable to modify in amplitude and phase the impulse response of the imaging system. This complex pupil function 15 is noted D(ξ, η), where ξ, η are the Cartesian coordinates in the plane 13 of the pupil 11.
A focusing defect is herein defined by the corresponding phase-shift Ψ(z, Z) of the optical wave at the edge of the pupil 11 with respect to the wave front of the conjugated point A′0. The pupil has a circular shape centred to the optical axis 12 and of minimum radius R. By definition:
      Ψ    ⁢                  ⁢          (              z        ,        Z            )        =                              π          ⁢                                          ⁢                      R            2                          λ            ⁢              (                              1            z                    +                      1            Z                    -                      1            F                          )              =                            2          ⁢          π                λ            ⁢              w        20            
where w20 represents the normal spacing, expressed in number of wavelengths, with respect to the reference sphere at the edge of the pupil due to the focusing defect.
In the Fresnel approximation, the expression of the complex amplitude of the electromagnetic field in the image space is given by the equation (2):
      U    ⁡          (              Z        ,        x        ,        y        ,        z            )        ∝                    1                              λ            2                    ·          zZ                    ·              e                  i          ⁢                                    k              ·                              (                                                      x                    2                                    +                                      y                    2                                                  )                                                    2              ⁢              z                                          ·              e                  i          ⁢                                    k              ·                              (                                                      X                    2                                    +                                      Y                    2                                                  )                                                    2              ⁢              Z                                            ⁢                  ×                  ∫          ∫                    pupil        ⁢                  D        ⁡                  (                      ξ            ,            η                    )                    ·              e                  i          ⁢                                                    k                ·                                  (                                                            ξ                      2                                        +                                          η                      2                                                        )                                            2                        ·                          (                                                1                  Z                                +                                  1                  z                                -                                  1                  F                                            )                                          ·              e                  -                      ik            ⁡                          [                                                                    (                                                                  X                        Z                                            +                                              x                        z                                                              )                                    ·                  ξ                                +                                                      (                                                                  Y                        Z                                            +                                              y                        z                                                              )                                    ·                  η                                            ]                                          ·      d        ⁢                  ⁢          ξ      ·      d        ⁢                  ⁢    η  
It is searched to determine a spatial filter of the phase-mask type for extending the focusing depth or, equivalently, extending the depth of field of an imaging system. Expressed mathematically, this problem amounts to define in the pupil plane a phase profile D(ζ, η) such that, for a given object point, the optical field near the conjugate image point has a property of invariance along the optical axis:
                                                        ∂                              |                                  U                  ⁡                                      (                                          x                      ,                      y                      ,                      z                                        )                                                  ⁢                                  |                  2                                                                    ∂              z                                =                      0            ⁢                          ∀              x                                      ,                              y            ⁢                                                  ⁢            for            ⁢                                                  ⁢            any            ⁢                                                  ⁢            z                    ∈          PM                                    (        3        )            
Generally, the equation (3) has no analytic solution.
The problem mentioned hereinabove may be formulated more simply in the case of an optical system with a symmetry of revolution and for an object point A located in the field of the system. A system of cylindrical coordinates is introduced and, in the pupil plane, the reduced radius is denoted ρ, defined by
      ρ    =                                        ξ            2                    +                      η            2                              R        ,with R the maximum radius of the pupil.
The equation (2) is hence reformulated:
      U    ⁡          (              Z        ,        r        ,        z            )        =                    2        ⁢        π        ⁢                                  ⁢                  R          2                                      λ          2                ·        zZ              ⁢          e              i        ⁢                              k            ·                          r              2                                            2            ⁢            z                                ⁢                  ∫        0        1            ⁢                                    D            ⁡                          (              ρ              )                                ·                      e                          i              ⁢                                                          ⁢                                                Ψ                  ⁡                                      (                                          z                      ,                      Z                                        )                                                  ·                                  ρ                  2                                                              ·                                    J              0                        ⁡                          (                                                kR                  z                                ·                r                ·                ρ                            )                                ·          ρ          ·                                          ⁢          d                ⁢                                  ⁢        ρ            
The normalized intensity is expressed as:
      I    ⁡          (              Z        ,        r        ,        z            )        =            1              I        0              |                  ∫                  0          ⁢          Δ                1            ⁢                                    D            ⁡                          (              ρ              )                                ·                      e                          i              ⁢                                                          ⁢                                                Ψ                  ⁡                                      (                                          z                      ,                      Z                                        )                                                  ·                                  ρ                  2                                                              ·                                    J              0                        ⁡                          (                                                kR                  z                                ·                r                ·                ρ                            )                                ·          ρ          ·                                          ⁢          d                ⁢                                  ⁢        ρ              ⁢          |      2      where I0 represents the intensity on the axis, limited by the diffraction, in the sensing plane of the imaging system with no spatial filter in infinite-focus conjugation.
In the case of an optical system that is not centred or that has such a numerical aperture that the Fresnel approximation does not apply, other models of distribution of the intensity I(Z, x, y, z) exist, which are well known by the one skilled in the art (see, for example, the Extended Nijboer-Zernike, ENZ, theory).
The publication J. Ojeda-Castaneda et al., “Ambiguity Function as a Design Tool for high focal depth”, Appl. Opt., Vol. 27, No. 4, 1988 describes an apodizer-type spatial filter, which modifies the amplitude of the pupil to modify the impulse response. More precisely, this publication compares the performances of the different apodizer filters allowing a stabilization of the response for a focusing defect of the order of 1λ, which corresponds to |Ψ|≤6.28. These amplitude apodizer filters have for main drawback to induce an attenuation of the luminosity of the imaging system and hence to strongly reduce the photometric efficiency of the imaging system.
Another publication describes a phase mask of polynomial profile, which allows stabilizing the point spread function (PSF) according to Ψ. The publication Cathey, E. Dowski, New Paradigm for Imaging System, Appl. Opt., Vol. 41, 2002, proposes, within the framework of the contemplated approximations, a phase mask having a polynomial spatial distribution of phase of order 3 in the plane of the pupil, or cubic phase mask, and introducing a defect of amplitude higher than 20 radians, which allows obtaining a defocusing range evaluated to |Ψ|≤30, i.e. a defocusing gain of a factor 10. Nevertheless, such cubic phase masks require a deconvolution of the sensed image by the resulting point spread function (PSF). These results have subsequently been extended to logarithmic profiles having analogous performances.
This type of phase mask has another drawback. Although the impulse response is almost-invariant in shape, the position of the barycentre of the impulse response is offset along the longitudinal axis. This offset produces a significant distortion of the image that cannot be corrected, even digitally, because it depends on the three-dimensional position of each object point. Finally, these masks are generally expensive to produce, because they must generally be performed by diamond machining.
The publication H. Wang et al., “High Focal Depth with a Pure-Phase Apodizer”, Appl. Optics, Vol. 40, No. 31, 2001, proposes a phase mask consisted of three concentric rings having binary phase-shifting between 0 and π. This phase mask, of simple manufacturing, produces no image distortion. The application of the diffraction theory allows calculating the distribution of intensity along the longitudinal optical axis of the optical system, for different values of the normalized radius a and b of the two internal rings. By a trial and error approach, the determination of optimized values of a and b allows making the axial distribution of intensity I(z, x=0, y=0) uniform and extending the depth of field. However, this method does not allow evaluating the distribution of intensity in the whole image field, and hence does not guarantee optimum performances in terms of spatial resolution. This type of phase mask finds applications in confocal beam-scanning microscopy, where a point by point image is formed. Combined with a post-processing of the image, such a mask allows extending the focusing defect to |Ψ|≤7.6 i.e. a defocusing gain of a factor two.
The use of such a ring phase mask has two limitations: on the one hand, the performance in terms of extension of the depth of field and/or of focusing depth remains limited, and on the other hand, in imaging, the criterion of invariance of the intensity is not sufficient to obtain an image of quality over the whole image field. Finally, the step of deconvolution requires an additional step in the chain of processing of the image.
Moreover, there also exist digital methods of image processing for allowing increasing the spatial resolution or the contrast of an imaging system. However, these methods involve a digital post-processing that is generally incompatible with the direct use (with no deconvolution) of the imaging system. These methods have limited performances.
Hence, a need exists for a method of designing a spatial filter making it possible to modify (extend or reduce) the depth of field and/or the focusing depth of an imaging system while ensuring an invariance of the image quality over this modified depth of field and/or modified focusing depth. Preferably, the imaging system allows obtaining an image on the area of modified depth of field and/or modified focusing depth, with no image post-processing.
In particular, it is desirable to extend the range of the focusing defects tolerated for an existing imaging system, so as to facilitate the manufacturing and the focusing settings of this imaging system and to extend the depth of field and/or the focusing depth of this imaging system without introducing a distortion or a degradation of the image quality.
More precisely, it is desirable to extend the depth of field of an imaging system with no image post-processing to obtain a depth of field extension at least as great as in the prior systems requiring a post-processing. Moreover, it is desirable to significantly extend the depth of field of an imaging system combining a spatial filter and a post-processing by comparison with the results obtained in the prior art systems.
There also exists a need for a spatial filter intended to be combined with an imaging system, the spatial filter being easy to manufacture, making it possible to modify the field and/or focusing depth of the imaging system, with no degradation of the image quality along the longitudinal optical axis of the optical system.